Trigonometric integral

Plot of the hyperbolic sine integral function Template:Math in the complex plane from Template:Math to Template:Math

Template:Math (blue) and Template:Math (green) shown on the same plot.

Sine integral in the complex plane, plotted with a variant of domain coloring.

Template:AnchorCosine integral in the complex plane. Note the branch cut along the negative real axis.

In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.

Sine integral

Plot of Template:Math for Template:Math.

Plot of the cosine integral function Template:Math in the complex plane from Template:Math to Template:Math

The different sine integral definitions are $$\mathrm{Si}(x)=\underset{0}{\overset{x}{\int }}\frac{\sin t}{t}dt$$ $$\mathrm{si}(x)=-\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{\sin t}{t}dt.$$

Note that the integrand $\frac{\sin (t)}{t}$ is the sinc function, and also the zeroth spherical Bessel function. Since Template:Math is an even entire function (holomorphic over the entire complex plane), Template:Math is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.

By definition, Template:Math is the antiderivative of Template:Math whose value is zero at Template:Math, and Template:Math is the antiderivative whose value is zero at Template:Math. Their difference is given by the Dirichlet integral, $$\mathrm{Si}(x)-\mathrm{si}(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin t}{t}dt=\frac{\pi }{2}\text{ or }\mathrm{Si}(x)=\frac{\pi }{2}+\mathrm{si}(x).$$

In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing if using a truncated sinc filter as a low-pass filter.

Related is the Gibbs phenomenon: If the sine integral is considered as the convolution of the sinc function with the Heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.

Cosine integral

Plot of Template:Math for Template:Math

The different cosine integral definitions are $$\mathrm{Cin}(x)\equiv \underset{0}{\overset{x}{\int }}\frac{1-\cos t}{t}\mathrm{d}t.$$

Template:Math is an even, entire function. For that reason, some texts define Template:Math as the primary function, and derive Template:Math in terms of Template:Math

$$\mathrm{Ci}(x)\equiv -\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{\cos t}{t}\mathrm{d}t$$ $$=\gamma +\ln x-\underset{0}{\overset{x}{\int }}\frac{1-\cos t}{t}\mathrm{d}t$$

$$=\gamma +\ln x-\mathrm{Cin}x$$ for $|\mathrm{Arg}(x)|<\pi ,$ where Template:Math is the Euler–Mascheroni constant. Some texts use Template:Math instead of Template:Math. The restriction on Template:Math is to avoid a discontinuity (shown as the orange vs blue area on the left half of the plot above) that arises because of a branch cut in the standard logarithm function (Template:Math).

Template:Math is the antiderivative of Template:Math (which vanishes as $x\to \mathrm{\infty }$). The two definitions are related by $$\mathrm{Ci}(x)=\gamma +\ln x-\mathrm{Cin}(x).$$

Hyperbolic sine integral

The hyperbolic sine integral is defined as $$\mathrm{Shi}(x)=\underset{0}{\overset{x}{\int }}\frac{\sinh (t)}{t}dt.$$

It is related to the ordinary sine integral by $$\mathrm{Si}(ix)=i\mathrm{Shi}(x).$$

Hyperbolic cosine integral

The hyperbolic cosine integral is

Plot of the hyperbolic cosine integral function Template:Math in the complex plane from Template:Math to Template:Math

$$\mathrm{Chi}(x)=\gamma +\ln x+\underset{0}{\overset{x}{\int }}\frac{\cosh t-1}{t}dt\text{ for }|\mathrm{Arg}(x)|<\pi ,$$ where $\gamma$ is the Euler–Mascheroni constant.

It has the series expansion $$\mathrm{Chi}(x)=\gamma +\ln (x)+\frac{x^2}{4}+\frac{x^4}{96}+\frac{x^6}{4320}+\frac{x^8}{322560}+\frac{x^{10}}{36288000}+O(x^{12}).$$

Auxiliary functions

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" $$\begin{array}{ccccccc}f(x)& \equiv & \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (t)}{t+x}dt& =& \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-xt}}{t^2+1}dt& =& \mathrm{Ci}(x)\sin (x)+[\frac{\pi }{2}-\mathrm{Si}(x)]\cos (x),\\ g(x)& \equiv & \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (t)}{t+x}dt& =& \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t{e}^{-xt}}{t^2+1}dt& =& -\mathrm{Ci}(x)\cos (x)+[\frac{\pi }{2}-\mathrm{Si}(x)]\sin (x).\end{array}$$ Using these functions, the trigonometric integrals may be re-expressed as (cf. Abramowitz & Stegun, p. 232) $$\begin{array}{ccc}\frac{\pi }{2}-\mathrm{Si}(x)=-\mathrm{si}(x)& =& f(x)\cos (x)+g(x)\sin (x),\text{ and }\\ \mathrm{Ci}(x)& =& f(x)\sin (x)-g(x)\cos (x).\end{array}$$

Nielsen's spiral

Nielsen's spiral.

The spiral formed by parametric plot of Template:Math is known as Nielsen's spiral. $$x(t)=a\times \mathrm{ci}(t)$$ $$y(t)=a\times \mathrm{si}(t)$$

The spiral is closely related to the Fresnel integrals and the Euler spiral. Nielsen's spiral has applications in vision processing, road and track construction and other areas.

Expansion

Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.

Asymptotic series (for large argument)

$$\mathrm{Si}(x)\sim \frac{\pi }{2}-\frac{\cos x}{x}(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots )-\frac{\sin x}{x}(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots )$$ $$\mathrm{Ci}(x)\sim \frac{\sin x}{x}(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots )-\frac{\cos x}{x}(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots ).$$

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at Template:Math.

Convergent series

$$\mathrm{Si}(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^3}{3!\cdot 3}+\frac{x^5}{5!\cdot 5}-\frac{x^7}{7!\cdot 7}\pm \cdots$$ $$\mathrm{Ci}(x)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n}}{2n(2n)!}=\gamma +\ln x-\frac{x^2}{2!\cdot 2}+\frac{x^4}{4!\cdot 4}\mp \cdots$$

These series are convergent at any complex Template:Mvar, although for Template:Math, the series will converge slowly initially, requiring many terms for high precision.

Derivation of series expansion

From the Maclaurin series expansion of sine: $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\cdots$$ $$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+\frac{x^8}{9!}-\frac{x^{10}}{11!}+\cdots$$ $$\therefore \int \frac{\sin x}{x}dx=x-\frac{x^3}{3!\cdot 3}+\frac{x^5}{5!\cdot 5}-\frac{x^7}{7!\cdot 7}+\frac{x^9}{9!\cdot 9}-\frac{x^{11}}{11!\cdot 11}+\cdots$$

Relation with the exponential integral of imaginary argument

The function $${\mathrm{E}}_1(z)=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\exp (-zt)}{t}dt\text{ for }\mathrm{\Re }(z)\ge 0$$ is called the exponential integral. It is closely related to Template:Math and Template:Math, $${\mathrm{E}}_1(ix)=i(-\frac{\pi }{2}+\mathrm{Si}(x))-\mathrm{Ci}(x)=i\mathrm{si}(x)-\mathrm{ci}(x)\text{ for }x>0.$$

As each respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to (Outside this range, additional terms which are integer factors of Template:Math appear in the expression.)

Cases of imaginary argument of the generalized integro-exponential function are $$\underset{1}{\overset{\mathrm{\infty }}{\int }}\cos (ax)\frac{\ln x}{x}dx=-\frac{{\pi }^2}{24}+\gamma (\frac{\gamma }{2}+\ln a)+\frac{{\ln }^{2}a}{2}+\sum _{n\ge 1}\frac{(-a^2{)}^n}{(2n)!(2n{)}^2},$$ which is the real part of $$\underset{1}{\overset{\mathrm{\infty }}{\int }}{e}^{iax}\frac{\ln x}{x}dx=-\frac{{\pi }^2}{24}+\gamma (\frac{\gamma }{2}+\ln a)+\frac{{\ln }^{2}a}{2}-\frac{\pi }{2}i(\gamma +\ln a)+\sum _{n\ge 1}\frac{(ia{)}^n}{n!n^2}.$$

Similarly $$\underset{1}{\overset{\mathrm{\infty }}{\int }}{e}^{iax}\frac{\ln x}{x^2}dx=1+ia[-\frac{{\pi }^2}{24}+\gamma (\frac{\gamma }{2}+\ln a-1)+\frac{{\ln }^{2}a}{2}-\ln a+1]+\frac{\pi a}{2}(\gamma +\ln a-1)+\sum _{n\ge 1}\frac{(ia{)}^{n+1}}{(n+1)!n^2}.$$

Efficient evaluation

Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015), are accurate to better than Template:Math for Template:Math, $$\begin{array}{ccc}\mathrm{Si}(x)& \approx & x\cdot \left(\frac{\begin{array}{c}1-4.54393409816329991\cdot 10^{-2}\cdot x^2+1.15457225751016682\cdot 10^{-3}\cdot x^4-1.41018536821330254\cdot 10^{-5}\cdot x^6\\ +9.43280809438713025\cdot 10^{-8}\cdot x^8-3.53201978997168357\cdot 10^{-10}\cdot x^{10}+7.08240282274875911\cdot 10^{-13}\cdot x^{12}\\ -6.05338212010422477\cdot 10^{-16}\cdot x^{14}\end{array}}{\begin{array}{c}1+1.01162145739225565\cdot 10^{-2}\cdot x^2+4.99175116169755106\cdot 10^{-5}\cdot x^4+1.55654986308745614\cdot 10^{-7}\cdot x^6\\ +3.28067571055789734\cdot 10^{-10}\cdot x^8+4.5049097575386581\cdot 10^{-13}\cdot x^{10}+3.21107051193712168\cdot 10^{-16}\cdot x^{12}\end{array}}\right)\\ & & \\ \mathrm{Ci}(x)& \approx & \gamma +\ln (x)+\\ & & x^2\cdot \left(\frac{\begin{array}{c}-0.25+7.51851524438898291\cdot 10^{-3}\cdot x^2-1.27528342240267686\cdot 10^{-4}\cdot x^4+1.05297363846239184\cdot 10^{-6}\cdot x^6\\ -4.68889508144848019\cdot 10^{-9}\cdot x^8+1.06480802891189243\cdot 10^{-11}\cdot x^{10}-9.93728488857585407\cdot 10^{-15}\cdot x^{12}\end{array}}{\begin{array}{c}1+1.1592605689110735\cdot 10^{-2}\cdot x^2+6.72126800814254432\cdot 10^{-5}\cdot x^4+2.55533277086129636\cdot 10^{-7}\cdot x^6\\ +6.97071295760958946\cdot 10^{-10}\cdot x^8+1.38536352772778619\cdot 10^{-12}\cdot x^{10}+1.89106054713059759\cdot 10^{-15}\cdot x^{12}\\ +1.39759616731376855\cdot 10^{-18}\cdot x^{14}\end{array}}\right)\end{array}$$

The integrals may be evaluated indirectly via auxiliary functions $f(x)$ and $g(x)$, which are defined by

$$\mathrm{Si}(x)=\frac{\pi }{2}-f(x)\cos (x)-g(x)\sin (x)$$		$$\mathrm{Ci}(x)=f(x)\sin (x)-g(x)\cos (x)$$
or equivalently
$$f(x)\equiv [\frac{\pi }{2}-\mathrm{Si}(x)]\cos (x)+\mathrm{Ci}(x)\sin (x)$$		$$g(x)\equiv [\frac{\pi }{2}-\mathrm{Si}(x)]\sin (x)-\mathrm{Ci}(x)\cos (x)$$

For $x\ge 4$ the Padé rational functions given below approximate $f(x)$ and $g(x)$ with error less than 10⁻¹⁶:

$$\begin{array}{ccc}f(x)& \approx & \frac{1}{x}\cdot \left(\frac{\begin{array}{c}1+7.44437068161936700618\cdot 10^2\cdot x^{-2}+1.96396372895146869801\cdot 10^5\cdot x^{-4}+2.37750310125431834034\cdot 10^7\cdot x^{-6}\\ +1.43073403821274636888\cdot 10^9\cdot x^{-8}+4.33736238870432522765\cdot 10^{10}\cdot x^{-10}+6.40533830574022022911\cdot 10^{11}\cdot x^{-12}\\ +4.20968180571076940208\cdot 10^{12}\cdot x^{-14}+1.00795182980368574617\cdot 10^{13}\cdot x^{-16}+4.94816688199951963482\cdot 10^{12}\cdot x^{-18}\\ -4.94701168645415959931\cdot 10^{11}\cdot x^{-20}\end{array}}{\begin{array}{c}1+7.46437068161927678031\cdot 10^2\cdot x^{-2}+1.97865247031583951450\cdot 10^5\cdot x^{-4}+2.41535670165126845144\cdot 10^7\cdot x^{-6}\\ +1.47478952192985464958\cdot 10^9\cdot x^{-8}+4.58595115847765779830\cdot 10^{10}\cdot x^{-10}+7.08501308149515401563\cdot 10^{11}\cdot x^{-12}\\ +5.06084464593475076774\cdot 10^{12}\cdot x^{-14}+1.43468549171581016479\cdot 10^{13}\cdot x^{-16}+1.11535493509914254097\cdot 10^{13}\cdot x^{-18}\end{array}}\right)\\ & & \\ g(x)& \approx & \frac{1}{x^2}\cdot \left(\frac{\begin{array}{c}1+8.1359520115168615\cdot 10^2\cdot x^{-2}+2.35239181626478200\cdot 10^5\cdot x^{-4}+3.12557570795778731\cdot 10^7\cdot x^{-6}\\ +2.06297595146763354\cdot 10^9\cdot x^{-8}+6.83052205423625007\cdot 10^{10}\cdot x^{-10}+1.09049528450362786\cdot 10^{12}\cdot x^{-12}\\ +7.57664583257834349\cdot 10^{12}\cdot x^{-14}+1.81004487464664575\cdot 10^{13}\cdot x^{-16}+6.43291613143049485\cdot 10^{12}\cdot x^{-18}\\ -1.36517137670871689\cdot 10^{12}\cdot x^{-20}\end{array}}{\begin{array}{c}1+8.19595201151451564\cdot 10^2\cdot x^{-2}+2.40036752835578777\cdot 10^5\cdot x^{-4}+3.26026661647090822\cdot 10^7\cdot x^{-6}\\ +2.23355543278099360\cdot 10^9\cdot x^{-8}+7.87465017341829930\cdot 10^{10}\cdot x^{-10}+1.39866710696414565\cdot 10^{12}\cdot x^{-12}\\ +1.17164723371736605\cdot 10^{13}\cdot x^{-14}+4.01839087307656620\cdot 10^{13}\cdot x^{-16}+3.99653257887490811\cdot 10^{13}\cdot x^{-18}\end{array}}\right)\end{array}$$

See also

• Logarithmic integral
• Tanc function
• Tanhc function
• Sinhc function
• Coshc function

References

• Template:AS ref

Further reading

• Template:Cite arXiv
• Template:Cite book
• Template:Cite web
• Template:Dlmf

External links

• http://mathworld.wolfram.com/SineIntegral.html
• Template:Springer
• Template:Springer